The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit $1$ as an eigenvalue. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space $(W,\overline{\Omega })$, having chosen a given reference...

Let $𝔄=\left(\begin{array}{cc}𝒜& ℳ\\ & ℬ\end{array}\right)$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜,ℬ)$-bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏,𝔄]\subseteq 𝔏$. A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$, every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$.

The notion of a $J^3$-triple is studied in connection with a geometrical approach to the generalized Hurwitz problem for quadratic or bilinear forms. Some properties are obtained, generalizing those derived earlier by the present authors for the Hurwitz maps S × V → V. In particular, the dependence of each scalar product involved on the symmetry or antisymmetry is discussed as well as the configurations depending on various choices of the metric tensors of scalar products of the basis elements. Then...

We obtain upper bounds for generalized indices of matrices in the class of nearly reducible Boolean matrices and in the class of critically reducible Boolean matrices, and prove that these bounds are the best possible.

Let $\parallel ·\parallel$ be a norm on the algebra ${ℳ}_n$ of all $n\times n$ matrices over $ℂ$. An interesting problem in matrix theory is that “Are there two norms ${\parallel ·\parallel }_1$ and ${\parallel ·\parallel }_2$ on ${ℂ}^n$ such that $\parallel A\parallel =max\{\parallel Ax{\parallel }_2\parallel x{\parallel }_1=1\}$ for all $A\in {ℳ}_n$?” We will investigate this problem and its various aspects and will discuss some conditions under which ${\parallel ·\parallel }_1={\parallel ·\parallel }_2$.

In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.

In this paper, we discuss the scheduling of a wide class of transportation systems. In particular, we derive an algorithm to generate a regular schedule by using max-plus algebra. Inputs of this algorithm are a graph representing the road network of public transportation systems and the number of public vehicles in each route. The graph has to be strongly connected, which means there is a path from any vertex to every vertex. Let us remark that the algorithm is general in the sense that we can allocate...

We introduce a new family of generalized Schröder matrices from the Riordan arrays which are obtained by counting of the weighted lattice paths with steps $E=(1,0)$, $D=(1,1)$, $N=(0,1)$, and $D^{\text{'}}=(1,2)$ and not going above the line $y=x$. We also consider the half of the generalized Delannoy matrix which is derived from the enumeration of these lattice paths with no restrictions. Correlations between these matrices are considered. By way of illustration, we give several examples of Riordan arrays of combinatorial interest. In addition,...